Existence of zero-order meromorphic solutions in detecting 𝑞-difference Painlevé equations

Korhonen, Risto; Wen, Zhi-Tao

doi:10.1090/tran/6491

Existence of zero-order meromorphic solutions in detecting $q$-difference Painlevé equations
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by Risto Korhonen and Zhi-Tao Wen PDF

Trans. Amer. Math. Soc. 368 (2016), 4993-5008 Request permission

Abstract:

The existence of zero-order meromorphic solutions is used as a sufficient condition in detecting $q$-difference equations of Painlevé type. It is shown that demanding the existence of at least one non-rational zero-order meromorphic solution $w(z)$ is sufficient to reduce a canonical class of $q$-difference equations with rational coefficients into a short list of Painlevé type $q$-difference equations, unless $w(z)$ is simultaneously a solution of a $q$-difference Riccati equation of a specific form.

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